A parallel is simply a long piece of hardened steel with a rectangular cross section. It is very accurately made, in particular, both pairs of its opposite sides are parallel.
Fig. assorted parallels 690
A parallel can be used as a flat surface. As such it can be used to test if two surfaces or a surface and a line or two lines are in the same plane. If is placed on any of the above pairs it will wobble if they are not both in the same plane
Where a flat surface, i.e., the parallel, is placed on a round surface, ie a round bar, it will only touch along its length a line that is parallel to the axis of the round surface.
A parallel can be used to test if two round parts of the same diameter are in the same plane. Place the parallel so it touches both at roughly the same angle. If they are not in the same plane then the parallel will wobble. The wider the parallel is, the better, in this case.
In some cases a small but very flat plate can be used to do the same thing. This can simply be a piece of gauge plate, say, 5mm thick and about 50mm square.
The use of parallels is covered elsewhere
MEW 13 p34